Mathematical Principles of Theoretical Physics

76 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

Usually, (2.4.37) is taken as an infinitesimal transformation as follows

Ω=I+iθaτa, θaare infinitesimal.

In this case, (2.4.37) can be written as

(2.4.38)

Ψ ̃=Ψ+iθaτaΨ,

G ̃aμ=Gaμ−λbcaθbGcμ−^1

g

∂μθa,

m ̃a=m+iθa(τam−mτa),

whereλbcaare the structure constants as defined in (2.4.26).
We have seen that the Dirac equations (2.4.30)-(2.4.32) are covariant under theSU(N)
gauge transformations (2.4.37) or (2.4.38).
What remains to do is to find out the action of the gauge fieldsGaμ, which is invariant
under both the Lorentz transformation and theSU(N)gauge transformation (2.4.37).
Recall that in (2.4.13), theU( 1 )gauge field action (2.4.15) is derived by using the com-
mutator of the covariant derivative operator:

[

Dμ,Dν

]

=DμDν−DνDμ.

We now derive in the same fashion theSU(N)action, called the Yang-Mills action.
By (2.4.34),
D ̃μ(D ̃νΨ ̃) =D ̃μ(ΩDνΨ) =Ω(DμDνΨ).

It follows that

(2.4.39)

[

D ̃μ,D ̃ν

]

Ψ ̃=Ω[Dμ,Dν]Ψ.

On the other hand, by (2.4.32), we have

i

g

[

Dμ,Dν

]

=

i

g

(∂μ+igGaμτa)(∂ν+igGaντa)−

i
g
(2.4.40) (∂ν+igGaντa)(∂μ+igGaμτa).

Notice that
∂μ∂ν=∂ν∂μ, ∂ν(Gaμτa) =∂νGaβτa+Gaβτa∂ν.

Then (2.4.40) becomes

i

g

[

Dμ,Dν

]

=(∂μGaν−∂νGaμ)τa−ig

[

Gaμτa,Gbντb

]

(2.4.41)

=(by( 2. 4. 26 ))

=(∂μGaν−∂νGaμ+gλbcaGbμGcν)τa.

Therefore we define

(2.4.42) Fμ νa =∂μGaν−∂νGaμ+gλbcaGbμGcν.